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In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory. Intuitively, subharmonic functions are related to convex functions of one variable as follows.
Subharmonic frequencies are frequencies below the fundamental frequency of an oscillator in a ratio of 1/ n, with n a positive integer. For example, if the fundamental frequency of an oscillator is 440 Hz, sub-harmonics include 220 Hz ( 1 ⁄ 2 ), ~146.6 Hz ( 1 ⁄ 3 ) and 110 Hz ( 1 ⁄ 4 ).
That is, f is harmonic on every complex line. A function f: U → R ∪ { − ∞} is plurisubharmonic, sometimes plush or psh for short, if it is upper-semicontinuous and for every a, b ∈ Cn, the function of one variable ξ ↦ f(a + bξ) is subharmonic (whenever a + bξ ∈ U ).
One can see that at some parameter values and initial conditions, the system’s oscillation spectrum is heavily contributed (almost dominated) by the \(3^{\text {rd }}\) subharmonic, i.e. the Fourier component of frequency \(\omega / 3 \approx \omega_{0}\).
Suppose that w(z0) = v1(z0). Then v1(z) w(z) w(z0) = v1(z0): Thus v1(z) is constant. Now either w(z) = v1(z) in a neighbourhood of z0 or w(z0) = v2(z0). But then v2(z) is also constant in a neighbourhood of z0 and either way it follows that w(z) is constant in a neighbourhood of z0.
23 de ene. de 2021 · Subharmonic function. A function $ u = u ( x): D \rightarrow [ - \infty , \infty ) $ of the points $ x = ( x _ {1} \dots x _ {n} ) $ of a Euclidean space $ \mathbf R ^ {n} $, $ n \geq 2 $, defined in a domain $ D \subset \mathbf R ^ {n} $ and possessing the following properties: 1) $ u ( x) $ is upper semi-continuous in $ D $; 2) for ...
A Closer Look for Vocalists. Learn exactly what subharmonic singing is, how to hear it in yourself and others, and when it occurs for vocalizers of all genres, voice types, and disciplines.
